Question: Ricardo throws a stone off a bridge into a river below. The stone's height (in meters above the water), $x$ seconds after Ricardo threw it, is modeled by $w(x)=-5(x-8)(x+4)$ What is the maximum height that the stone will reach?
Solution: The stone's height is modeled by a quadratic function, whose graph is a parabola. The maximum height is reached at the vertex. So in order to find the maximum height, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $w(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} w(x)&=0 \\\\ -5(x-8)(x+4)&=0 \\\\ \swarrow &\searrow \\\\ x-8=0\text{ or }&x+4=0 \\\\ x={8}\text{ or }&x={-4} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({8})+({-4})}{2}=\dfrac42= 2$ The vertex's $x$ -coordinate is $ 2$. Now let's find $w({2})$ : $\begin{aligned} w( 2)&=-5( 2-8)( 2+4) \\\\ &=-5(-6)(6) \\\\ &=180 \end{aligned}$ In conclusion, the maximum height that the stone will reach is $180$ meters.